Linear Spaces
Fields
Definition: A field is a set F of scalars with two operations, addition and multiplication, such that the following axioms hold:
Addition
- (A1) a+b=b+a for all a,b∈F (commutativity of addition)
- (A2) (a+b)+c=a+(b+c) for all a,b,c∈F (associativity of addition)
- (A3) There exists an element 0∈F such that a+0=a for all a∈F (existence of additive identity)
This element is commonly denoted by 0F​.
- (A4) For every a∈F there exists an element −a∈F such that a+(−a)=0F​ (existence of additive inverse)
This element is commonly denoted by (−a).
Multiplication
- (M1) a⋅b=b⋅a for all a,b∈F (commutativity of multiplication)
- (M2) (a⋅b)⋅c=a⋅(b⋅c) for all a,b,c∈F (associativity of multiplication)
- (M3) There exists an element in F such that a⋅1F​=a for all a∈F (existence of multiplicative identity)
- (M4) For every a∈F except 0F​ there exists an element a−1∈F such that a⋅a−1=1F​ (existence of multiplicative inverse)
Distributivity
- (D1) a⋅(b+c)=a⋅b+a⋅c for all a,b,c∈F (distributivity of multiplication over addition)
At least two elements must exist in a field, additive identity 0F​ and multiplicative identity 1F​.
Field examples include R, C, Q, Zp​ where p is a prime number. R are the real numbers, C are the complex numbers, Q are the rational numbers, Zp​ are the integers modulo p.
In a sense, what I understand from the fields are, mathematical concepts that are used to define the operations on the elements of the vector spaces.
Linear Spaces
Fn
Definition: The set of all lists of length n with elements from a field F is denoted by Fn.
Fn={(x1​,x2​,…,xn​)∣xi​∈F,i=1,2,…,n}
Addition and scalar multiplication are defined as follows:
(x1​,x2​,…,xn​)+(y1​,y2​,…,yn​)=(x1​+y1​,x2​+y2​,…,xn​+yn​)
α(x1​,x2​,…,xn​)=(αx1​,αx2​,…,αxn​)
#EE501 - Linear Systems Theory at METU